A digital image is represented by values at specified grid locations. A gradient vector represents the maximum change of the image values. Since image variations are of broad interest, gradient operators are extensively used in many image-processing applications, such as image diffusion, edge detection and curvature computation.
Several gradient operators are commonly used today in the image-processing field. Examples of such gradient operators are: Simple Difference, Sobel, “Isotropic”, and Prewitt. Such operators and their applications are described in greater detail in U.S. Pat. Nos. 4,561,022; 5,870,495; 6,031,928; 6,345,107; 6,408,109; 6,480,615; 6,535,623; 6,788,826; and 7,027,658; US2002/0146163; and in published U.S. patent applications US2005/0152591; US2006/0062458; US2006/0072844; and US2006/0104495.
U.S. Pat. No. 4,561,022 uses a biased finite difference operator for reducing noise in television images. U.S. Pat. No. 5,870,495 uses a Sobel operator as a fuzzy gradient corrector for edge detection on images to recognized geometric shapes. U.S. Pat. No. 6,031,928 and U.S. Pat. No. 6,345,107 use a central finite difference operator for processing height data to obtain image data. U.S. Pat. No. 6,408,109 uses a Sobel operator for edge detection and for detection of sub-pixels in digital images.
U.S. Pat. No. 6,480,615 uses an adaptive biased finite difference operator for video segmentation to estimate motion within a sequence of data frames for optical flow. U.S. Pat. No. 6,535,623 uses a central finite difference operator for medical image processing, in particular for cardiac data from magnetic resonance image data, with segmentation and analysis of the medical data. U.S. Pat. No. 6,788,826 uses a central finite difference operator for performing digital filtering to correct artifacts in a digital image. U.S. Pat. No. 7,027,658 uses gradient operators for geometrically accurate data compression and decompression.
US 2002/0146163 uses a central finite difference operator for determining the shape of objects. US 2005/0152591 uses gradient Magnitude, Angle and Radial distance (MARS) filter for filtering medial image data. US 2006/0062458 uses a Sobel operator on gradient direction to perform object recognition identifying objects in an image. US 2006/0072844 reduces noise to perform gradient-based image restoration and enhancement. US 2006/0104495 performs image processing on medical image data for local visualization of tubular structures in the body such as blood vessels.
None of these patents and patent applications are applied to seismic data, and the methods applied by these patents and patent applications rely on known image data processing techniques which have deficiencies in that digital anisotropy is significant and is not reduced significantly or at all. The prior art fails to determine how to obtain a better or optimal gradient operator for improved image processing.
In addition, since all of these operators (Simple Difference, Sobel, “Isotropic”, and Prewitt) are generally implemented in the Cartesian coordinate system, which is intrinsically anisotropic (i.e., spacings between grid or lattice points are longer diagonally), an obvious question arises as to whether such operators are truly isotropic. In other words; if an original image rotates by a certain angle, the resultant gradient would not rotate by exactly the same angle.
In Jahne and Haubecher, “Computer Vision and Applications”, Academic Press, 1999, the numerical inaccuracy of two kinds of numerical gradient operators was analyzed by applying them to plane waves, and it was pointed out that the inaccuracy (the angle error and the magnitude anisotropy) depends on both the frequency and the direction of the plane waves. The calculations in this analysis showed that for a simple-difference operator, the angle error can be as much as 7 degrees and the magnitude varies 17% at half the Nyquist frequency.
Following from the analysis in Jahne and Haubecher, which involves applying numerical operators to plane waves with a Fourier transform, any image can be decomposed into a set of sinusoids or equivalently plane-wave functions. Therefore, an operator's properties on any image can then be studied by examining its properties on plane waves.
As an extension of Jahne and Haubecher's analysis, one object of the present invention is to provide an improved solution to the problem of reducing anisotropic inaccuracies.